Can I set my pose position like the new rest position?
Apply the armature modifier - armature in rest pose doesn't deform anything, so we need also a mesh to match the new rest pose.
Pose > Apply Pose as Rest Pose. This will set current pose as the rest pose. It sets only bones poses - constraints or custom properties in complicated or badly designed rigs might make the rig act weird with bones in different rest state!
Add the armature modifier back.